UNDER RECONSTRUCTION BELOW

  1. Note that all objects at a given radius from the center of mass orbit with approximately the same velocity. This is dictated by Newtonian physics. Thus, to measure this orbital velocity, we only need to measure the orbital velocity of tracer astronomical objects, usually stars collectively (NOT individually since they are NOT resolved) or molecular clouds or other forms of interstellar medium (ISM).

  2. The creator of Image 2 neglected to put any axis scales and units on the plot, but the x-axis would typically extend to 30 to 50 kpc and the vertical axis to 250 km/s since ∼ 200 km/s is the typical plateau orbital velocity of galaxy rotation curves.

  3. How galaxy rotation curves behave beyond ∼ 50 kpc is NOT well known empirically since we run out of observable tracers of the matter which is primarily dark matter.

  4. The naively expected rotation curve A is what the observed visible baryonic matter (mainly stars) (i.e., luminous matter) predicts.

    The observed rotation curve B tells us that there is dark matter that we see NO trace of except through its gravitational effects.

  5. The dark matter is ∼ 85 % of the matter in the observable universe, and thus is ∼ 5.5 times more abundant than baryonic matter (which is ordinary matter made of protons, neutrons, and electrons).

    However, the ratio of baryonic matter to dark matter in galaxies seems vary various ranging from 1/10??? to 1/30 to less (Dekel et al. 2019, Figure 1).

    Baryonic matter in the intergalactic medium (IGM) is ∼ 9 more abundant than baryonic matter in galaxies (see pie_chart_cosmic_energy.html). So most baryonic matter by far is NOT in galaxies. The intergalactic medium (IGM) is mainly primordial cosmic composition (fiducial values by mass fraction: 0.75 H, 0.25 He-4, 0.001 D, 0.0001 He-3, 10**(-9) Li-7). Of course, there is lots of dark matter in intergalactic space, but that is usually NOT counted as part of the intergalactic medium (IGM).

    Because of its dominance in galaxies, dark matter is the main determinant of the centers of mass of the galaxies.

  6. What is dark matter? Big Bang theory (which is a very well supported theory nowadays) is what tells (via Big Bang nucleosynthesis era (cosmic time ∼ 10--1200 s ≅ 0.17--20 m)) that dark matter is NOT baryonic matter.

    Moreover, baryonic matter is hard to hide in all respects. If the dark matter were baryonic matter, we believe we would have detected by some means by now.

    In fact, we do NOT know what dark matter is. It is thought most likely to be an exotic fundamental particle that we have NOT detected yet, except, as aforesaid, by its gravitational effects. This exotic fundamental particle (i.e., the dark matter particle) apparently only interacts very weakly with itself and baryonic matter, except through gravity.

    Another theory is that primordial black holes (PBHs) (i.e., those formed before Big Bang nucleosynthesis era (cosmic time ∼ 10--1200 s ≅ 0.17--20 m)) make up the dark matter.

    Hopefully, we will discover what dark matter is in the 2020s.

  7. It seems odd that we CANNOT tell the difference between dark matter particles (which are microscopic scale) and PBHs (which are usually assumed to be macroscopic scale), but we CANNOT. They both orbit chaotically around the center of mass of galaxies subject to gravitational interactions only in the case of PBHs and mainly in the case of dark matter particles.

  8. The distribution of the dark matter of a galaxy forms what is called a dark matter halo. Dark matter halos are probably mostly roughly spherically symmetric it is thought and extend out beyond the baryonic matter, but how far is hard to determine since we run out of observable tracers made of the baryonic matter as aforesaid.

  9. Now what is the explanation for the behavior of the galaxy rotation curves seen in Image 2?

    We can analyze the behavior qualitatively using a very simplified picture. There is a general formula for circular orbit orbital velocity (which is a uniform circular motion) that follows from Newtonian physics (specifically Newton's 3rd law of motion and Newton's law of universal gravitation) for a test particle in a spherically symmetric mass distribution:

      v = sqrt[GM(r)/r]  , 

    where the gravitational constant G = 6.67430(15)*10**(-11) (MKS units), r is radius from the center of the distribution, M(r) is the interior mass (i.e., mass interior to a sphere of radius r), and the mass exterior to radius r has absolutely NO effect as proven by the shell theorem. We consider special cases of formula:

    1. Falling orbital velocity case: or the Keplerian orbit case:
        v = sqrt[GM(R)/r] ∝ 1/sqrt(r)  , 

      where all the mass is confined to within R and r > R. This is the Keplerian orbit case which holds for planets, moons, etc.. This case would hold approximately for the outer part of galaxies if the baryonic matter were all the matter. This is case for curve A in Image 2.

    2. Constant orbital velocity case:
        v = sqrt[GCr/r]=sqrt[GC]  , 

      where C is a constant. In this case M(r)=Cr ∝ r and there is a cancellation of r's and v stays constant with r. This is the case for the plateau region of curve B in Image 2 and approximately the case of real galaxies showing that there is a lot of dark matter. It usually extends out as far as we can find observable tracers to measure orbital velocity from. Dark matter halos extend well beyond the baryonic matter which sits in their gravitational wells.

    3. Rising orbital velocity case:
         v = sqrt[G(4π/3)r**3*ρ/r]=sqrt[qrt[G(4π/3)ρ]*r ∝ r  , 

      where ρ is uniform density (i.e., constant density). For this case the velocity increasely linearly with r. This is very approximately the case for inside planets---if test particles were free to orbit there. The result implies that the gravitational force in planets increases linearly from their centers to their surface. On the size scale of a kiloparsec, galaxies at least to order of magnitude seem constant density in the central regions, and so this this velocity cases suggests the orbital velocities should go toward zero near the centers of galaxies. This behavior is seen for both curves A and B in Image 2.

    So we have explained qualitatively the galaxy rotation curves in Image 2.

    In fact, from analyzing galaxy rotation curves, the density profiles of galaxies can be determined to a degree. However, the outer regions of the dark matter halos are hard to determine because we run out of tracers for the orbital velocities. Also deviations from spherical symmetry in dark matter halos are also hard to detect.

  10. Note on very small scales near the center of galaxies, there can be high orbital velocities for orbits around central supermassive black holes (SMBHs). But those orbits are probably usually on a smaller radial scale than shown in Image 2.

    To illustrate: for Keplerian orbits around SMBHs, we have the following fiducial value formulae:

      v = (2.0739 ... km/s)*sqrt[M/(10**6*M_☉)/(r/ 1 kpc)]
      v = (65.58 ... km/s)*sqrt[M/(10**6*M_☉)/(r/ 1 pc)]
      v = (65.58 ... km/s)*sqrt[M/(10**9*M_☉)/(r/ 1 kpc)]
      v = (2073.9 ... km/s)*sqrt[M/(10**9*M_☉)/(r/ 1 pc)] 

    As one can see on the size scale of 1 kpc (which is about where the plateau orbital velocity starts in Image 2), even a very large SMBH of mass 10**9 M_☉ gives velocities (∼ 65 km/s) that are still relatively small compared to the 200 km/s of the plateau orbital velocity. However, it is possible in some galaxies with very massive central SMBHs (i.e., ∼> 10**9 M_☉) that galaxy rotation curves do stay very high (∼> 200 km/s) in the central region of the galaxies.

    alien_click_to_see_image click on image

  11. Image 2 Caption: A full explanation of the circular orbit orbital velocity inside and outside of a uniform density sphere along with a diagram and an orbital velocity versus radius plot.

    https://upload.wikimedia.org/wikipedia/commons/b/be/Grav_field_sphere.svg

  12. Image 3 Caption: To complement Image 2, Image 3 shows the gravitational field inside and outside of a uniform density sphere. Actually, the plotted curve does NOT seem quantitatively accurate. So take it as a cartoon.

    The formulae for the gravitational field are:

     g = G(4π/3)ρ*r**3/r**2 
       = G(4π/3)ρ*r 
       = (GM/R**2)(r/R) for inside,
     g = GM/R**2        at the surface,
     g = GM/r**2        for outside, 

    where the gravitational constant G = 6.67430(15)*10**(-11) (MKS units), ρ is the uniform density, and M is the total mass of the sphere. The formulae can be derived from the shell theorem. The gravitational field points radially inward.

    Note that the gravitational field goes to the zero at r = 0. Although, stars, planets, etc. are NOT uniform density spheres, they also have gravitational fields going to the zero at their centers. This can be proven by the shell theorem.

    Also note that outside of the sphere, the gravitational field obeys an inverse-square law as for a point mass. Thus, though this is NOT clear in Image 3, the gravitational field outside the sphere only goes to zero as →∞. This outside behavior also holds for any spherically symmetric mass distribution (as can be proven by the shell theorem), and so holds for stars, planets, etc.

File: Galaxies file: galaxy_rotation_1bb.html